Numerical Methods for Stochastic Differential Equations in Stiefel Manifolds via the Cayley Transform

Abstract

Stochastic differential equations evolving in a Stiefel manifold occur in several applications in Science and Engineering. For ordinary differential equations evolving in Stiefel manifolds there is a solid literature on numerical implementation guaranteeing adherence to the manifold. But for stochastic differential equations, numerical methods are in their infancy. Indeed some existing schemes fail to satisfy the required geometric constraints. We develop a new and efficient scheme to simulate a stochastic differential equation evolving in a Stiefel manifold, based on the Cayley transform. In particular, we show how to construct drift and diffusion terms to obey geometric conditions, ensuring evolution in the Stiefel manifold. Comparative simulations illustrate the new scheme showing that it is geometry preserving over large numbers of time steps.